Article

# Smooth solutions to the nonlinear wave equation can blow up on Cantor sets

03/2011;
Source: arXiv

ABSTRACT We construct $C^\infty$ solutions to the one-dimensional nonlinear wave
equation $$u_{tt} - u_{xx} - \tfrac{2(p+2)}{p^2} |u|^p u=0 \quad \text{with} \quad p>0$$ that blow up on any prescribed uniformly space-like $C^\infty$
hypersurface. As a corollary, we show that smooth solutions can blow up (at the
first instant) on an arbitrary compact set.
We also construct solutions that blow up on general space-like $C^k$
hypersurfaces, but only when $4/p$ is not an integer and $k > (3p+4)/p$.

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• ##### Chapter:Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
ninth Dover printing, tenth GPO printing ; Dover.
• ##### Article:Differentiability of the blow-up curve for one dimensional nonlinear wave equations
Archive for Rational Mechanics and Analysis 02/1985; 91(1):83-98. · 2.05 Impact Factor
• Source
##### Article:Blow-up Surfaces for Nonlinear Wave Equations, I
[show abstract] [hide abstract]
ABSTRACT: We introduce a systematic procedure for reducing nonlinear wave equations to characteristic problems of Fuchsian type. This reduction is combined with an existence theorem to produce solutions blowing up on a prescribed hypersurface. This first part develops the procedure on the example □u = exp(u); we find necessary and sufficient conditions for the existence of a solution of the form ln(2/2) + v, where { = 0} is the blow-up surface, and v is analytic. This gives a natural way of continuing solutions after blow-up.
Communications in Partial Differential Equations 01/1993; 18(3-4):431-452. · 0.89 Impact Factor

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### Keywords

$C^\infty$ solutions

first instant

one-dimensional nonlinear wave

smooth solutions

solutions